jmc

Question

Let a and b be the two real values of x for which[3]x + [3]20 - x = 2The smaller of the two values can be expressed as p - q, where p and q are integers. Compute p + q.

Accepted Answer

Let a=[3]x, b = [3]20-x. Then a+b = 2 and a^3 + b^3 = 20. Factoring,a^3 + b^3 = (a+b)((a+b)^2-3ab) = 2(4-3ab)= 8-6ab=20 ab = -2 Solving a+b=2, ab=-2 gives us the quadratic a^2 - 2a - 2 = 0. The quadratic formula yields a = 2 - 122 = 1 - 3, and x = a^3 = (1-3)^3 = 1 - 33 + 9 - 33 = 10 - 108. Therefore, p+q=118.